Free-by-cyclic graphical discreteness conjecture

Let $F_n$ be the free group of rank $n$, let $\phi$ be an automorphism, and consider the free-by-cyclic group

$$F_n\rtimes_{\phi}\mathbb{Z}.$$

Assume that this group is one-ended and hyperbolic, and has trivial JSJ decomposition. A group is graphically discrete when it has only trivial lattice envelopes.

Free-by-cyclic graphical discreteness conjecture. A one-ended, hyperbolic free-by-cyclic group $F_n\rtimes_{\phi}\mathbb{Z}$ with trivial JSJ decomposition is graphically discrete.

The supplied text presents this as an open question among the paper's conjectures and gives no known cases or resolution.